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Hopf bifurcation analysis of a predator-prey system with Holling type IV functional response and time delay. (English) Zbl 1187.34116
This paper is concerned with a predator-prey system with Holling type IV functional response and time delay. By choosing the delay as a bifurcation parameter, the local asymptotic stability of the positive equilibrium and existence of local Hopf bifurcations are analyzed. Based on the normal form and the center manifold theory, the formulas for determining the properties of Hopf bifurcation of the predator-prey system are derived. Finally, to support these theoretical results, some numerical simulations are given to illustrate the results.
MSC:
34K60Qualitative investigation and simulation of models
34K18Bifurcation theory of functional differential equations
34K19Invariant manifolds (functional-differential equations)
92D25Population dynamics (general)
References:
[1]Celik, C.: The stability and Hopf bifurcation for a predator – prey system with time delay, Chaos, solitons and fractals 37, 87-99 (2008) · Zbl 1152.34059 · doi:10.1016/j.chaos.2007.10.045
[2]Collings, J. B.: The effects of the functional response on the bifurcation behavior of a mite predator – prey interaction model, J. math. Biol. 36, 149-168 (1997) · Zbl 0890.92021 · doi:10.1007/s002850050095
[3]Hale, J.; Lunel, S. V.: Introduction to functional differential equations, (1993)
[4]Hassard, B.; Kazarinoff, N.; Wan, Y.: Theory and applications of Hopf bifurcation, (1981)
[5]Holling, C. S.: The functional response of predators to prey density and its role in mimicry and population regulation, Mem. entomol. Soc. can. 46, 1-60 (1965)
[6]Hsu, S. B.; Huang, T. W.: Global stability for a class of predator – prey systems, SIAM J. Appl. math. 55, 763-783 (1995) · Zbl 0832.34035 · doi:10.1137/S0036139993253201
[7]Hsu, S. B.; Huang, T. W.: Hopf bifurcation analysis for a predator – prey system of Holling and Leslie type, Taiwan J. Math. 3, 35-53 (1999) · Zbl 0935.34035
[8]Leslie, P. H.; Gower, J. C.: The properties of a stochastic model for the predator – prey type of interaction between two species, Biometrika 47, 219-234 (1960) · Zbl 0103.12502
[9]Li, Y.; Xiao, D.: Bifurcations of a predator – prey system of Holling and Leslie types, Chaos, solitons and fractals 34, 606-620 (2007) · Zbl 1156.34029 · doi:10.1016/j.chaos.2006.03.068
[10]Lin, G.; Hong, Y.: Delay induced oscillation in predator – prey system with beddington – deangelis functional response, Appl. math. Comput. 190, 1296-1311 (2007) · Zbl 1117.92055 · doi:10.1016/j.amc.2007.02.012
[11]May, R. M.: Stability and complexity in model ecosystems, (1973)
[12]Nindjin, A. F.; Aziz-Alaoui, M. A.; Cadivel, M.: Analysis of a predator – prey model with modified Leslie – gower and Holling-type II schemes with time delay, Nonlinear anal. 7, 1104-1118 (2006) · Zbl 1104.92065 · doi:10.1016/j.nonrwa.2005.10.003
[13]Lu, Z.; Liu, X.: Analysis of a predator – prey model with modified Holling – tanner functional response and time delay, Nonlinear. anal.: real world appl. 9, 641-650 (2008) · Zbl 1142.34053 · doi:10.1016/j.nonrwa.2006.12.016
[14]Sokol, W.; Howell, J. A.: Kinetics of phenol exidation by washed cells, Biotechnol. bioeng. 23, 2039-2049 (1980)
[15]Song, Y.; Yuan, S.: Bifurcations analysis in a predator – prey system with time delay, Nonlinear anal. 7, 265-284 (2006) · Zbl 1085.92052 · doi:10.1016/j.nonrwa.2005.03.002
[16]Xu, R.; Chaplain, M. A. J.: Persistence and global stability in a delayed predator – prey system with michaelis – menten type functional response, Appl. math. Comput. 130, 441-455 (2002) · Zbl 1030.34069 · doi:10.1016/S0096-3003(01)00111-4
[17]Yan, X.: Stability and Hopf bifurcation for a delayed prey – predator system with diffusion effects, Appl. math. Comput. 192, 552-566 (2007) · Zbl 1193.35098 · doi:10.1016/j.amc.2007.03.033